Lie Groups

研究人员提出了七个相互交织的研究项目，所有这些项目都是关于群论和几何之间的界面。在大多数情况下，这些建议是几何激励的方法，以解决现代调和分析中的特定问题，与半单李群的表示理论有关。第一步是构造和分析实约化李群的奇异么正表示，分两步进行：(I)几何构造这些表示，作为旗域上齐次向量丛的上同调的Frechet空间表示；(Ii)通过从旗域到其线性循环空间的双重纤维变换(复彭罗斯变换是特例)，将表示空间变换到由微分方程组定义的Stein流形上的泛函空间。第二个项目是构造和分析一类无限维李群的酉表示和其他表示，有限维李群的直接极限，特别是有限维实李群和复约化李群在解析范畴和代数范畴中的严格直接极限。第三个项目是通过综合相对Schwartz空间的结构分析，完成对一般半单李群的Harish-Chandra Schwartz空间的研究工作。这位研究人员的第四个项目是将狄拉克上同调的概念扩展到部分Dirac上同调的概念，以便它适用于Plancerel公式中出现的半单Lie群的所有表示。第五个项目是将研究者的等谱群方法从球面空间形式的设置推广到局部对称黎曼空间的设置。第六个项目是研究离散级数表示对一类具有重要几何意义的子群的某些限制。第七，研究人员将继续开发一种方法，用于从基本数据直接读取有限维实还原李群的可允许表示的特征和增长性质(渐近性)，这些基本数据指定了它们在旗域上齐次向量丛的上同调空间上的构造。这里的一个目标是以一种直接适用于第一个和第二个项目的方式来实现这一点。这七个研究项目都依赖于使用对称性来澄清分析(在一个情况下是几何)问题。传统上，对称性考虑被用来通过减少变量的数量来简化问题，但在这里，它们被用来允许使用几何和分析的洞察力、工具和结果。对称性体现在群论中，群论是对称性概念的代数抽象。但现代李群理论融合了经典分析(微积分、微分方程式……)并与几何(黎曼几何、辛几何、凯勒几何等)密切相关。群在其上充当对称性的构型。几何和分析综合的一个重要方面是量子化的几何形式，它特别适合于在几个项目中考虑的有限维群的种类。这种几何量子化最初受到物理学的启发，并由数学家详细地开发出来，反过来在数学和物理的各种环境中也非常有用。在七个项目中，有六个项目就是这种情况，几何量子化与现代微分几何相结合，以理解各种解析问题。这一点在第一和第四个项目中尤其明显，它们的目标可以被视为一种反量化。

AbstractWolfThe investigator proposes seven intertwined research projects, all on the interface between group theory and geometry. For the most part these proposals are geometrically motivated approaches to specific problems in modern harmonic analysis connected to the representation theory of semi-simple Lie groups. The first is an approach to construction and analysis of singular unitary representations of real reductive Lie groups, in two steps: (i) geometric construction of those representations, as Frechet space representations on cohomology of homogeneous vector bundles over flag domains, and (ii) transforming the representation space to a space offunctions on a Stein manifold defined by a system of differential equations,by means of a double fibration transform (the complex Penrose transform is a particular case) from the flag domain to its linear cycle space. The second project is an approach to construction and analysis of unitary and other representations for a class of infinite dimensional Lie groups, the direct limits of finite dimensional Lie groups, especially strict direct limits of finite dimensional real and complex reductive Lie groups, both in the analytic category and in the algebraic category. The third project is to complete the investigator's work on the Harish-Chandra Schwartz space of a general semisimple Lie group, by synthesizing structural analyses of the relative Schwartz spaces. The investigator's fourth project is to extend the notion of Dirac cohomology to a notion of partial Diraccohomology so that it applies to all the representations of a semisimpleLie group that appear in the Plancherel formula. The fifth project is toextend the investigator's isospectral group method from the setting of spherical space forms to the setting of locally symmetric Riemannian spaces.The sixth project is to investigate certain restrictions of discrete seriesrepresentations to a class of subgroups of great geometric interest. And,seventh, the investigator will continue his development of a method for direct reading of the character and growth properties (asymptotics) of admissible representations of finite dimensional real reductive Lie groups from the basic data that specify their construction on cohomology spaces of homogeneous vector bundles over flag domains. One goal here is to do this in such a way that is directly applicable to the first and second projects. These seven research projects all depend on the use of symmetry to clarifyanalytic (and in one case geometric) problems. Traditionally symmetryconsiderations are used to simplify matters by decreasing the number ofvariables, but here they are used to enable the use of insight, tools andresults from geometry and analysis. The symmetries are embodied in group theory, which is the algebraic abstraction of the notion of symmetry. But modern Lie group theory incorporates classical analysis (calculus, differential equations...) and is closely tied to the geometry (Riemannian, symplectic, Kaehler, ...) of the configurations on which the groups act as symmetries. An important aspect of this synthesis of geometry and analysis is a geometric form of quantization that is particularly well suited to the sorts of finite dimensional groups considered in several of the projects. This geometric quantization, originally inspired by physics and developed in some detail by mathematicians, has in turn been very useful in a variety of settings in mathematics and physics. This is very much the case in six of the seven projects, where the geometric quantization is combined with modern differential geometry to understand various analytic problems. This is especially evident in the first and fourth projects, whose objectives can be viewed as a sort of dequantization.